From: Joe Shipman <shipman at savera.com>
JS>One point which should be emphasized in the current discussion of Real
JS>Analysis as a primary motivator for foundational work: almost all "core
JS>mathematics" can be coded into second-order arithmetic, which is the
JS>foundation developed for Real Analysis.
This raises for me a fundamental question about the continuum. Clearly
it is both geometric (a topological space amenable to delta-epsilon
analysis etc.) and algebraic (a real closed field with an immensely
useful algebraic closure, the complex numbers). The foundational
question that springs to mind is, which comes first for the continuum,
geometry or algebra? Is the continuum of geometric origin, or algebraic?
One result possibly bearing on this is that the continuum is a
final coalgebra, dual to the natural numbers as an initial algebra.
This appears in "On Coalgebra of Real Numbers" by Dusko Pavlovic and
myself, presented at Amsterdam in the Proceedings of the Second Workshop
on Coalgebraic Methods in Computer Science (CMCS'99), Electronic Notes
in Theoretical Computer Science, Volume 19. Online it's the eighth paper
in
http://www.elsevier.nl:80/cas/tree/store/tcs/free/entcs/store/contents.htt?jrnl=tcs&sctn=entcs&mode=sub&vol=19
Better however is the longer journal version, with new title "The
continuum as a final coalgebra," to appear soon in TCS and currently
available at
http://boole.stanford.edu/pub/continuum.ps.gz
(Now that all the panic and confusion is long past I can look back on the
circumstances of the Amsterdam talk with some bemusement. I was giving
a weeklong series of presentations at CeBIT on our brand-new Matchbox PC,
described in
http://wearables.stanford.edu
CeBIT, in Hanover in late March, is a real zoo, 600,000 visitors spread
over 25 buildings on more than a hundred acres, about 3 times the size
of Comdex. In mid-CeBIT I took Sunday off to present the paper, taking
the train from Hanover to Amsterdam on Saturday night and staying with
friends of Dusko, who turned out to be an utterly charming couple heavily
into journalism who were overwhelmingly hospitable, drinking wine and
discussing technology with me until 2 am. The 9 am invited talk was by
Peter Wegner, who presented his arguments why God had also created the
continuum, scooping the slogan I'd been planning to use for our talk in
the afternoon.)
A better (in my view) functor than the ones Dusko and I used has recently
been found by Peter Freyd and discussed at some length on the categories
mailing list. Whereas the simpler of our functors was product with N
(natural numbers) in Set, Freyd's was what I call "fusion-squaring"
in Bip, the category of bipointed sets, the operation X v X (in Freyd's
ASCII notation) in which two copies of the two-constant algebra (X,0,1)
are made and the constant 1 of the first copy is then identified (fused)
with 0 of the second. Passing to lexicographic or ordinal product with
the ordinal omega in the category Pos of posets made our coalgebraic
structure explicit in the form of an actual final coalgebra instead of
merely definable by corecursion; the corresponding passage for Freyd is
to the category of posets with top and bottom.
Coalgebra would appear to offer one attractive formalization of the
intuitively obvious duality of algebra and geometry, whose roots lie in
Stone duality and which has over the past two decades been the subject of
many papers on duality in the universal algebra community, most notably
the long collaboration of the Australian algebraist Brian Davey and the
Oxford algebraist Hilary Priestley. (UA is only 30-50% as alienated
from category theory as is FOM, for, one guesses, the same basic reason:
category theory is in fact alien, as I acknowledged in my 11/1/97 response
on FOM to Peter Simpson's plaintive "I have never really understood the
way category theorists seem to view the rest of mathematics". It is
alien for much the same reason as the world Alice entered when she passed
through the looking-glass seemed alien: about half of your knowledge is
now backwards, but all still otherwise intact if you just look carefully
enough. I enthusiastically encourage all mathematicians imbued with any
spirit of adventure to attempt the passage through that daunting portal,
you will be richly rewarded if you make it through in one piece.)
My own involvement with duality started in 1991, leading to twenty-odd
papers, available at
http://chu.stanford.edu
initially on higher dimensional automata (as the solution to a puzzling
paradox in the duality of schedules and automata) and subsequently
on Chu spaces, a class of objects having the same homogeneity and
universal applicability as ZF sets while doing for structured objects
(normally understood heterogeneously via a vast zoo of categories) what
sets do for unstructured objects. I take the foundational importance of
duality seriously enough to have license plate DUELITY, the more usual
seule-entendre spelling being already taken.
JS>Manifolds, Hilbert Space, countable algebras, separable metric spaces,
JS>etc., may be more easily investigated in the full ZFC framework, but
JS>don't strictly require more of a foundation than Real Analysis does.
Evidence that at least some mathematicians think axiomatically about the
continuum, as opposed to encodings in second-order logic or first-order
ZFC, follows from the existence of finite geometries over finite fields,
and other applications of finite field theory. Would those ever so
useful finite fields have occurred to anyone if no one had thought to
view the continuum axiomatically?
Vaughan Pratt